We argue by contradiction. Consider a sequence of \(\Sigma _i \in \mathcal {S}_n^*(\Lambda -\varepsilon ) \setminus \mathcal {S}_n^\textnormal {gen}\)  and \(\mathcal {M}_i\)  ancient cyclic unit-regular integral Brakke flows in \(\mathbb {R}^{n+1}\)  with \(\lambda (\tilde{\mathcal {M}}_i)\le F(\Sigma )\)  so that \(\tilde{\mathcal {M}}_i\)  does not smoothly cross the flow \((-\infty ,0) \ni t\mapsto ^n\lfloor \sqrt{-t}\Sigma _i\)  and so that there are points \((\mathbf {x}_i,t_i) \in (\mathbb {R}^{n+1}\times \mathbb {R})\setminus (\mathbf {0},0)\)  with
\(\Theta _{\tilde{\mathcal {M}}_i}(\mathbf {x}_i,t_i) \ge F(\Sigma _i) - o(1)\) 
as \(i\rightarrow \infty \) . We can assume that \(|(\mathbf {x}_i,t_i)| = 1\) .
By Lemma REF  and Allard's theorem [1]}, [2]}, we can pass to a subsequence so that \(\Sigma _i\)  converges in \(C^\infty _\textrm {loc}\)  to \(\Sigma \in \mathcal {S}_n(\Lambda )\) . By Brakke's theorem [3]}, [4]}, \(\Sigma \)  is non-flat. Because cylinders are isolated in \(C^\infty _\textrm {loc}\)  by [5]}, we thus see that \(\Sigma \in \mathcal {S}_n^*(\Lambda ) \setminus \mathcal {S}_n^\textnormal {gen}\) . Note that \(F(\Sigma _{i})\rightarrow F(\Sigma )\) .
We now pass to a further subsequence so that \((\mathbf {x}_i,t_i) \rightarrow (\mathbf {x}_0,t_0) \in \mathbb {R}^{n+1}\times \mathbb {R}\)  with \(|(\mathbf {x}_0,t_0)| = 1\)  and the Brakke flows \(\tilde{\mathcal {M}}_i\)  converge to an ancient cyclic unit-regular integral Brakke flow \(\tilde{\mathcal {M}}\)  with \(\lambda (\tilde{\mathcal {M}}) \le F(\Sigma )\) . By upper semi-continuity of Gaussian density, (REF ) implies that \(\Theta _{\tilde{\mathcal {M}}}(\mathbf {x}_0,t_0) \ge F(\Sigma )\) . Beause \(\lambda (\tilde{\mathcal {M}}) \le F(\Sigma )\) , \(\tilde{\mathcal {M}}\)  is a self-similar flow around \((\mathbf {x}_0,t_0)\) . By stability of smoothly crossing flows, Lemma REF , \(\tilde{\mathcal {M}}\)  does not smoothly cross \((-\infty ,0) \ni t\mapsto ^n\lfloor \sqrt{-t}\Sigma \) .
Consider any tangent flow to \(\tilde{\mathcal {M}}\)  at \(t=-\infty \) . By Huisken's monotonicity formula and Lemma REF  there is a smooth shrinker \(\tilde{\Sigma }\)  so that this tangent flow at \(t=-\infty \)  corresponds to some \(\tilde{\Sigma }\in \mathcal {S}_{n}(\Lambda )\)  with multiplicity-one. By the Frankel property for self-shrinkers (cf. [6]}) and the strong maximum principle, if \(\tilde{\Sigma }\ne \Sigma \)  then the flows \(t\mapsto ^n\lfloor \sqrt{-t}\Sigma \)  and \(t\mapsto ^n\lfloor \sqrt{-t} \tilde{\Sigma }\)  smoothly cross each other at some point. This contradicts stability of smooth crossings.
We conclude that any tangent flow to \(\tilde{\mathcal {M}}\)  at \(t=-\infty \)  is the flow associated to \(\Sigma \) . Since \(\tilde{\mathcal {M}}\)  is self-similar around \((\mathbf {x}_{0},t_{0})\)  we find
\(\tilde{\mathcal {M}}(t) = ^n\lfloor (\sqrt{t_0 - t} \Sigma + \mathbf {x}_0)\) 
for \(t < t_{0}\) . Since \(\tilde{\mathcal {M}}\)  does not smoothly cross \(t\mapsto ^n\lfloor \sqrt{-t}\Sigma \) , we see that there is an open set \(\Omega \subset \mathbb {R}^{n+1}\)  with \(\partial \Omega = \Sigma \)  so that
\(\sqrt{t_0 - t} \Sigma + \mathbf {x}_0 \subset \sqrt{-t} \, \bar{\Omega }\) 
for \(t < \min \lbrace 0,t_{0}\rbrace \) . We can thus apply Proposition REF  to conclude that (up to a rotation) \(\Sigma = \hat{\Sigma }\times \mathbb {R}\)  for \(\hat{\Sigma }\in \mathcal {S}_{n-1}^*(\Lambda )\) . By hypothesis \((\dagger \dagger _{n,\Lambda })\) , \(\hat{\Sigma }\in \mathcal {S}^\textrm {gen}_{n-1}\)  so \(\Sigma = \hat{\Sigma }\times \mathbb {R}\in \mathcal {S}^\textrm {gen}_n\) . This is a contradiction.
